Spacelike foliations by $(n−1)$-umbilical hypersurfaces in spacetimes

Abstract

We consider the problem of whether a given spacetime admits a foliation by $(n−1)$-umbilical spacelike hypersurfaces. We introduce the notion of a timelike closed partially conformal vector field in a spacetime and show that the existence of a vector field of this kind guarantees in turn the existence of that foliation. We then construct explicit examples of families of $(n−1)$-umbilical spacelike hypersurfaces in the de Sitter space. Imposing the further condition of having constant $r$-th mean curvature, we give the complete description of any leaf of a foliation of the de Sitter space by these hypersurfaces. Finally, in a spacetime foliated by $(n−1)$-umbilical spacelike hypersurfaces we characterize the immersed spacelike hypersurfaces which are $(n−1)$-umbilical.